/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 86 Write the domain in interval not... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 86

Write the domain in interval notation. $$ k(x)=\log _{3}(5 x+6) $$

Short Answer

Expert verified
The domain in interval notation is \((-\frac{6}{5}, \infty)\).

Step by step solution

01

- Identify the condition for the logarithm

The argument of the logarithm must always be greater than 0. For the function \(k(x) = \log_{3}(5x + 6)\), the argument is \(5x + 6\). Set up the inequality \(5x + 6 > 0\).
02

- Solve the inequality

Solve the inequality \(5x + 6 > 0\). Subtract 6 from both sides to obtain \(5x > -6\). Divide both sides by 5 to isolate \(x\), resulting in \(x > -\frac{6}{5}\).
03

- Write the domain in interval notation

The solution \(x > -\frac{6}{5}\) can be written in interval notation as \((-\frac{6}{5}, \infty)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithmic functions
A logarithmic function is a mathematical function that helps us determine the exponent or power to which a base number must be raised to get another number. You often see logarithms with bases like 10 (common logarithm) or e (natural logarithm), but they can have any positive number as their base, except 1.

In the case of the function k(x) = log鈧�(5x + 6), the base is 3. This function shows us the exponent to which we must raise 3 to obtain the expression 5x + 6. It's important to remember that the argument of the logarithm, which is the part inside the log, must always be positive. So, for 5x + 6 to be valid, it must be greater than 0.

This unique requirement of logarithms plays a crucial role in determining the domain of logarithmic functions.
inequalities
Inequalities are mathematical statements that show that two expressions are not equal and there's a relationship of greater than or less than between them. You may see symbols like > (greater than) or < (less than) used in inequalities. They are essential in solving problems where we're interested in a range of values rather than specific numbers.

For our function k(x) = log鈧�(5x + 6), we need to solve the inequality 5x + 6 > 0 to find when the argument of the logarithm is valid. Here are the steps:
  • Subtract 6 from both sides to get 5x > -6.
  • Then, divide both sides by 5 to isolate x, resulting in x > -6/5.
By solving this inequality, we've found that the domain of our function includes all x values that are greater than -6/5.
interval notation
Interval notation is a way of representing a range of values along a number line. This method uses brackets and parentheses to show which numbers are included or excluded from the interval. For example:
  • (a, b) means all numbers between a and b, but not including a and b.
  • [a, b] means all numbers between a and b, including a and b.
  • (a, b] means all numbers between a and b, including b but not a.
  • [a, b) means all numbers between a and b, including a but not b.

For the function k(x) = log鈧�(5x + 6), we've determined that x > -6/5. In interval notation, this is written as (-6/5, 鈭�). This tells us that x can be any number greater than -6/5, extending up to infinity.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Fluorodeoxyglucose is a derivative of glucose that contains the radionuclide fluorine- \(18\left({ }^{18} \mathrm{~F}\right) .\) A patient is given a sample of this material containing \(300 \mathrm{MBq}\) of \({ }^{18} \mathrm{~F}\) (a megabecquerel is a unit of radioactivity). The patient then undergoes a PET scan (positron emission tomography) to detect areas of metabolic activity indicative of cancer. After \(174 \mathrm{~min}\), one-third of the original dose remains in the body. a. Write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the radioactivity level \(Q(t)\) of fluorine- 18 at a time \(t\) minutes after the initial dose. b. What is the half-life of \({ }^{18} \mathrm{~F}\) ?

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{5} 3 $$

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{5}\left(\frac{1}{x}\right)=\frac{1}{\log _{5} x} $$

Find the difference quotient \(\frac{f(x+h)-f(x)}{h} .\) Write the answers in factored form. $$f(x)=2^{x}$$

A function of the form \(P(t)=a b^{t}\) represents the population of the given country \(t\) years after January 1,2000 . a. Write an equivalent function using base \(e\); that is, write a function of the form \(P(t)=P_{0} e^{k t} .\) Also, determine the population of each country for the year 2000 . $$\begin{array}{|l|c|c|c|} \hline \text { Country } & P(t)=a b^{t} & P(t)=P_{0} e^{k t} & \begin{array}{c} \text { Population } \\ \text { in } 2000 \end{array} \\ \hline \text { Haiti } & P(t)=8.5(1.0158)^{t} & & \\ \hline \text { Sweden } & P(t)=9.0(1.0048)^{t} & & \\ \hline \end{array}$$ b. The population of the two given countries is very close for the year 2000 , but their growth rates are different. Determine the year during which the population of each country will reach 10.5 million. c. Haiti had fewer people in the year 2000 than Sweden. Why did Haiti reach a population of 10.5 million sooner?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.